Delay calculation method capable of calculating delay time with small margin of error

ABSTRACT

A delay calculation method that is capable of calculating delay time with a small margin of error is provided for delay calculation in a logic circuit. The operating characteristics of transistor are expressed with a fixed resistance and a power supply voltage that changes with time. The power supply voltage is represented as a waveform which is a combination of two straight lines: the one indicating that the voltage, after a fixed delay of t 0 , increases to V 1  during Δt 1 ; and the one indicating that the voltage increases from V 1  to E during Δt 2  and thereafter remains at the fixed value of E. A difference in the shapes of input waveforms is adopted as a correction parameter to determine the values of Δt 1 , V 1 , and Δt 2.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a delay calculation method forcalculating delay time of, for example, each macrocell with transistors.

2. Description of the Background Art

For timing verification of semiconductor integrated circuits (SOC:System on Chip) equipped with large-scale systems, delay time in eachinstance must be calculated with accuracy. One of well-knownhigh-precision delay calculation methods is that using circuitsimulations such as SPICE. In circuit simulations, the analysis ofoperating points of each device is made on matrix calculation. Thiscalculation is conducted at each lapse of a very short time to obtain avoltage value at every node. In this technique, a huge matrix must beanalyzed for the analysis of operation at a certain time, so an increasein the number of devices will markedly increase the amount ofcalculation. Further, the necessity of conducting matrix calculation ateach lapse of a very short time makes impossible to conduct an analysiswithin a realistic time, for semiconductor integrated circuits whichinclude an extremely large number of devices such as large-scale SOCs.To solve this problem, another delay calculation method using a delaycalculator has been employed.

The delay calculator previously adds an input waveform generatingcircuit and an output load respectively to the input and output of everymacrocell (logic circuit) that can cause a delay, and it conduct circuitsimulations (SPICE) using a plurality of kinds (usually 10 to 100) ofinput waveforms and output load capacitances. At this time, every inputwaveform is replaced and represented as time (a Tslew value) requiredfor transition from low to high state (or high to low state).

From simulation results obtained here, parameters which are necessaryfor calculating delay time as a function of the Tslew value and theoutput load capacitance are extracted and stored in a delay-parameterdatabase. This is done for each path when each macrocell has a pluralityof different changing paths.

Using the delay-parameter database created in this way, the delaycalculator performs delay calculations on a SOC. First, delay time inthe instance where the Tslew value is set by default or by the designeris calculated using the delay-parameter database. For delay calculation,information on the output load is necessary, and it can be produced inadvance from layout results obtained with a layout extractor. By sodoing, delay time and an output waveform in that instance can bedetermined. Since the Tslew value to be inputted in the followinginstance can be determined from the output waveform, the delay time andthe output waveform for the following instance can be determined usingthis Tslew value. This operation is performed for all instances, wherebyany delay time in the SOC can be calculated.

For example, Japanese Patent Application Laid-open No. 2002-123568 hasdisclosed one of the delay calculation methods for calculating delaytime only from Tslew values and output load information.

However, in real SOCs, depending on the topology of a circuit or RCnetwork to be configured, there are various shapes of input waveformsthat differ from the one used to create the delay-parameter database.That is, there exist a lot of input waveforms that have the same Tslewvalue but have different waveform shapes.

In conventional delay calculation methods, delay time is calculated onlyfrom Tslew values and output load information. Thus, in the case wherethe output load capacitance is the same and each input waveform has thesame Tslew value, the same delay time is calculated for different shapesof input waveforms.

Consequently, the results of delay calculations in conventional delaycalculation methods have a margin of error of around 8 percent even ifparameters are optimized somehow. At present, SOCs become faster, so amargin of error in delay calculation must be kept at 5 percent at most.From this, the 8-percent inherent margin of error has made circuitdesign difficult.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a delay calculationmethod for calculating delay time with a small margin of error.

The present invention is intended for a delay calculation method for alogic circuit including a MOS transistor, the transistor being modeledusing a resistance element whose resistance value is fixed and a powersupply whose voltage varies with time. A shape of a voltage waveform ofthe power supply is determined according to input waveform informationthat determines the shape of an input waveform inputted to the logiccircuit.

Since the shape of the voltage waveform of the power supply which modelsthe transistor is determined according to parameters that determine theshape of the input waveform, delay time can be calculated with a smallmargin of error.

These and other objects, features, aspects and advantages of the presentinvention will become more apparent from the following detaileddescription of the present invention when taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the sampling of circuit connection information according toa preferred embodiment of the present invention;

FIG. 2 illustrates the structure of circuit configuration information;

FIG. 3 is a schematic diagram of a load model;

FIG. 4 is a schematic diagram of an inverter;

FIG. 5 shows current waveform patterns;

FIGS. 6 and 7 are characteristic curves showing the change in powersupply voltage with time;

FIG. 8 shows the Ids-Vds characteristic curve of an E(t) model;

FIG. 9 shows a characteristic curve defining the relationship betweengate voltage and saturation current;

FIG. 10 shows an example of actual input waveforms along with itsapproximate line;

FIG. 11 is a diagram for explaining an operating point;

FIG. 12 is a flow chart for explaining a delay calculation methodaccording to the preferred embodiment of the present invention; and

FIGS. 13 to 16 show the output waveform by the delay calculation methodaccording to the preferred embodiment of the present invention, and theoutput waveform by SPICE.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Like conventional delay calculation methods, a delay calculation methodaccording to a preferred embodiment of the present invention inputs aTslew value of an input waveform, delay parameters for calculating delaytime as a function of the Tslew value and an output load capacitance,and output load information. The delay calculation method of thispreferred embodiment further inputs input waveform information thatgives the shapes of input waveforms. The method then outputs a delaytime, the following input waveform, and the Tslew value of the followinginput waveform.

FIGS. 1 to 3 shows the processing flow of delay calculation according tothis preferred embodiment. FIG. 1 shows a configuration of circuitconnection, for example when inverters 1 and 2 are connected through aninterconnect line 3. FIG. 4 shows the structure of the inverter 1. Asshown in FIG. 4, the inverter 1 is formed of a PMOS transistor P1 and anNMOS transistor N1. A power supply voltage Vdd is applied to the sourceof the PMOS transistor P1. The drain of the PMOS transistor P1 isconnected to the drain of the NMOS transistor N1 and to an outputterminal Y1. The source of the NMOS transistor N1 is grounded. The gatesof the PMOS and NMOS transistors P1 and N1 are connected to an inputterminal A1. The structure of the inverter 2 is identical and thus notdescribed here.

As shown in FIG. 2, circuit configuration information is generated basedon circuit connection information shown in FIG. 1. The circuitconfiguration information consists of a source model 6 corresponding tothe inverter 1, and a load component 8. The source model 6 includes apower supply 4 and a resistor 5, in which the resistor 5 has its one endconnected to the power supply 4 and its other end connected to one endof an RC distributed constant circuit 7 forming the load component 8.

The load component 8 includes the RC distributed constant circuit 7corresponding to the interconnect line 3, and an input pin capacitor Cg.The RC distributed constant circuit 7 has its one end connected to thesource model 6 and its other end connected to one end of the input pincapacitor Cg. The other end of the input pin capacitor Cg is grounded.

Next, as shown in FIG. 3, a load model 9 which is equivalent to thecircuit configuration information is generated. The load model 9consists of capacitors (capacitances) C1 and C2, which are respectivelyoutput and input capacitances of the RC distributed constant circuit 7,and a resistor (resistance) R of the RC distributed constant circuit 7in the load component 8. One end of the resistor R is connected to oneend of the input capacitor C2 and to the other end of the resistor 5 ofthe source model 6. The other end of the input capacitor C2 is grounded.The other end of the resistor R is connected to one end of the outputcapacitor C1, and the other end of the output capacitor C1 is grounded.

According to the present invention, the power supply 4 of theabove-configured source model 6 is designed to have time dependency(hereinafter, the magnitude of voltage at the power supply 4 isindicated by E(t)). Then, a model (referred to as an “E(t) model”) thatrepresents transistor operation when an input waveform is received isconstructed. That is, the E(t) model is a model that determines theshape of the waveform of E(t) (hereinafter referred to as the “E(t)waveform”) so that the output current i(t) of the source model 6 flowingto the load model 9 reproduces the waveform of the drain current Ids ofthe transistors forming the inverter 1. Thus, the shape of the E(t)waveform is determined by taking into account the waveform of the draincurrent Ids of the transistors.

Now, the waveform of the drain current Ids will be described. Thefollowing description is for the case where the input waveform (inputsignal) that transitions from high to low is inputted from the inputterminal A1. The shape of the input waveform is assumed to linearlychange from high (potential Vdd) to low (potential of 0 V).

At this time, the potential of the output terminal Y1 of the inverter 1transitions from low to high, because the PMOS transistor P1 of theinverter 1 transitions from off to on and thereby the output load ischarged.

FIG. 5 is a diagram for explaining the waveform of the drain current Idsthrough the PMOS transistor P1 when an input waveform that transitionsfrom high to low is inputted to the input terminal A1 of the inverter 1.The horizontal axis shows the source-drain voltage (drain voltage) Vds,and the vertical axis shows the drain current Ids. Dashed lines showIds-Vds curves of a transistor for different magnitudes of thegate-source voltage (gate voltage) Vgs.

As shown in FIG. 5, depending on the drive capability of each macrocelland the output load capacitance, the current waveform has two patterns:a first pattern in which region I where output current increases withtime is followed by region II where output current decreases graduallyand is then followed by region III where output current decreaseslinearly; and a second pattern in which region I where output currentincreases with time is immediately followed by region III where outputcurrent decreases rapidly and linearly.

The current waveform is described in more detail below. In the initialstate, the gate voltage Vgs of the PMOS transistor P1 is 0 V, and sincethe output load is not charged, the drain voltage Vds equals Vdd.

First, we describe the case where the output load capacitance is large,i.e., where the degree of decrease in the drain voltage Vds with thecharging of the output load is smaller than the degree of change in thegate voltage Vgs. As the input signal given to the input terminal A1 ofthe inverter 1 transitions from high to low, the gate voltage Vgsincreases and the drain current Ids starts to flow through the PMOStransistor P1.

The magnitude of the drain current Ids increases with increasing gatevoltage Vgs. Also, the drain voltage Vds drops from Vdd with thecharging of the output load with the drain current Ids. Since the largerthe output load capacitance the smaller the degree of decrease in thedrain voltage Vds, the PMOS transistor P1 goes into its saturationregion when the gate voltage Vgs reaches Vdd (region I). Region I ishereinafter also referred to as the “saturation region with varying gatevoltage” or “first region where current changes with varying gatevoltage”.

Then, the drain voltage Vds decreases gradually with the charging of theoutput load, whereby the drain current Ids decreases gradually while themagnitude of the gate voltage Vgs is constant (region II). Region II ishereinafter also referred to as the “saturation region with fixed gatevoltage” or “second region where current decreases gradually while thegate voltage is constant”.

Then, when the output load is further charged and the drain voltage Vdsdecreases to a certain degree, the PMOS transistor P1 goes into itslinear region where the drain current Ids decreases linearly. When thecharging of the output load is complete, the drain voltage Vds becomes 0V and the drain current Ids stops flowing (region III). Region III ishereinafter also referred to as the “linear region” or “third regionwhere current decreases while the gate voltage is constant”.

As above described, when the output load capacitance is large, thecurrent waveform is as shown by the first pattern in FIG. 5.

Next, we describe the case where the output load capacitance is small,i.e., where the degree of decrease in the drain voltage Vds with thecharging of the output load is greater than the degree of change in thegate voltage Vgs. When the output load capacitance is somewhat small,region II will disappear. For example, consider the case where, when thegate voltage Vgs reaches Vdd, the output load is charged to a certainextent and the transistor goes into its linear region. In this case, thedrain current Ids increases toward region III as shown in FIG. 5 withincreasing gate voltage Vgs (region I), and then rapidly decreases afterthe gate voltage Vgs reaches Vdd (region III).

When the output load capacitance becomes further smaller, the transistormoves from region I to region III before the gate voltage Vgs reachesVdd. The second pattern shown in FIG. 5 is such a pattern that theoutput load capacitance is small and thus, the transistor moves fromregion I to region III before the gate voltage Vgs reaches Vdd.

Next, we discuss the E(t) waveform that reproduces the current waveformdescribed above. Firstly discussed is the first pattern of the currentwaveform. On the line of the first pattern, region III is a linearregion where the current flowing is determined by the time constant of acircuit, so that E(t)=E(=Vdd). Since the source model 6 representstransistor operation by E(t) and the value of the resistor 5, andcurrent in region III decreases linearly with decreasing drain voltageVds, a current waveform that decreases linearly with time can bereproduced when E(t)=E.

In regions I and II, different waveforms of E(t) are applied. Forrespective regions I and II, E(t) is represented by different functionsof time (however, both the functions are continuous at a point of timecorresponding to regions I and II). In this preferred embodiment, forease of calculation, regions I and II are represented by continuousstraight lines with different inclinations as shown in FIG. 6.

More specifically, region I corresponds to a region where the draincurrent Ids increases rapidly. To represent this drain current Ids inthe source model 6, region I is brought into correspondence with astraight line with a high degree of inclination with respect to time.

Region II corresponds to a region where the drain current Ids degreasesgradually while the magnitude of the gate voltage Vgs is constant atVdd. To reproduce this gradually decreasing drain current Ids in thesource model 6, region II must be brought into correspondence with astraight line indicating that voltage increases with time because thevoltage applied to one end of the resistor 5 increases with the chargingof the output load capacitance. If voltage is constant with respect totime, the rate of decrease in the drain current Ids becomes too high.

The time when E(t) starts to increase does not necessarily coincide withtime t=0. The PMOS transistor P1 has a threshold voltage Vth, andcurrent starts to flow when the input waveform meets and exceeds thethreshold voltage Vth. Thus, E(t) starts to increase when the inputwaveform meets and exceeds the threshold voltage Vth. This duration oftime is referred to as a fixed delay t0. Assuming that the time when theinput waveform has finished its transition to 0 or Vdd is the Tslewvalue, the fixed delay t0 can be represented as a function of the Tslewvalue and can be approximated for example by the equation:t0(Tslew)=K10+K20·Tslew^(α). In this case, K10, K20, α are delayparameters which are stored in the delay-parameter database as functioninformation concerning an input slew rate that determines the fixeddelay t0.

According to this preferred embodiment, as shown in FIG. 6, the E(t)waveform is plotted as a line graph, showing that, after a lapse of thefixed delay t0, E(t) increases first to V1 during Δt1 and then to Eduring Δt2, where t0, Δt1, V1, and Δt2 are the parameters.

In the case of the second pattern, region II does not exist. In thiscase, assuming that Δt2=0 and V1=E, the E(t) waveform is such thatregion I directly moves to region III (FIG. 7).

When using the E(t) waveform shown in FIG. 6, the waveform of an outputterminal voltage V2(t) (c.f. FIG. 3) during the rise of the E(t)waveform can be expressed as follows.

If functions ƒ and α are defined as:

$\begin{matrix}\left\{ \begin{matrix}{{f\left( {t,{\Delta\; t},V} \right)} = {\begin{Bmatrix}{t - \frac{z}{p_{1}p_{2}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{1}}t} \right)}} -} \\{\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{2}}t} \right)}}\end{Bmatrix}\frac{V}{\Delta\; t}}} \\{{a\left( {t,V} \right)} = {\left\{ {1 - {\frac{p_{1} - z}{p_{1} - p_{2}}{\exp\left( {{- p_{1}}t} \right)}} + {\frac{p_{1} - z}{p_{1} - p_{2}}{\exp\left( {{- p_{2}}t} \right)}}} \right\} V}}\end{matrix} \right. & (1) \\{where} & \; \\\left\{ \begin{matrix}{z = {\frac{1}{{RC}\; 1} + \frac{1}{{RC}\; 2}}} \\{p_{1},{p_{2} = {\frac{1}{2}\begin{Bmatrix}{\left( {\frac{1}{{RC}\; 1} + \frac{1}{{RC}\; 2} + \frac{1}{{RsC}\; 2}} \right) \pm} \\\sqrt{\left( {\frac{1}{{RC}\; 1} + \frac{1}{{RC}\; 2} + \frac{1}{{RsC}\; 2}} \right)^{2} - \frac{4}{{RRsC}\; 1C\; 2}}\end{Bmatrix}}}}\end{matrix} \right. & (2)\end{matrix}$a model waveform of V2(t) can be expressed as follows:when Δt1=0,

$\begin{matrix}{{V\; 2(t)} = \left\{ \begin{matrix}0 & \left( {0 \leq t \leq {t\; 0}} \right) \\{{a\left( {{t - {t\; 0}},{V\; 1}} \right)} + {f\left( {{t - {t\; 0}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)}} & \left( {0 < t \leq {{t\; 0} + {\Delta\; t\; 2}}} \right) \\\begin{matrix}{{a\left( {{t - {t\; 0}},{V\; 1}} \right)} + {f\left( {{t - {t\; 0}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)} -} \\{f\left( {t,{{{- t}\; 0} - {\Delta\; t\; 2}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)}\end{matrix} & \left( {t > {{t\; 0} + {\Delta\; t\; 2}}} \right)\end{matrix} \right.} & (3)\end{matrix}$and when Δt2=0,

$\begin{matrix}{{V\; 2(t)} = \left\{ \begin{matrix}0 & \left( {0 \leq t \leq {t\; 0}} \right) \\{f\left( {{t - {t\; 0}},{\Delta\; t\; 1},E} \right)} & \left( {0 < t \leq {{t\; 0} + {\Delta\; t\; 1}}} \right) \\{{f\left( {{t - {t\; 0}},{\Delta\; t\; 1},E} \right)} + {f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 1},E} \right)}} & \left( {t > {{t\; 0} + {\Delta\; t\; 1}}} \right)\end{matrix} \right.} & (4)\end{matrix}$and when Δt1≠0 and Δt2≠0,

$\begin{matrix}{{V\; 2( t)} = \left\{ \begin{matrix}0 & \left( {0 \leq t \leq {t\; 0}} \right) \\{f\left( {{t - {t\; 0}},{\Delta\; t\; 1},{V\; 1}} \right)} & \left( {0 < t \leq {{t\; 0} + {\Delta\; t\; 1}}} \right) \\\begin{matrix}{{f\left( {{t - {t\; 0}},{\Delta\; t\; 1},{V\; 1}} \right)} -} \\{{f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 1},{V\; 1}} \right)} +}\end{matrix} & \left( {{{t\; 0} + {\Delta\; t\; 1}} < t \leq {{t\; 0} +}} \right. \\{f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)} & \left. {{\Delta\; t\; 1} + {\Delta\; t\; 2}} \right) \\\begin{matrix}{{f\left( {{t - {t0}},{\Delta\; t\; 1},{V\; 1}} \right)} -} \\{{f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 1},{V\; 1}} \right)} +} \\{{f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)} -} \\{f\left( {{t - {t\; 0} - {\Delta\; t\; 1}},{\Delta\; t\; 2},{E - {V\; 1}}} \right)}\end{matrix} & \left( {t > {{t\; 0} + {\Delta\; t\; 1} + {\Delta\; t\; 2}}} \right)\end{matrix} \right.} & (5)\end{matrix}$

The procedures for determining the values Δt1, V1, and Δt2 are describedbelow. Looking at the shifts of operating points of the transistor shownin FIG. 5, there are three boundary points from region I to region II,from region I to region III, and from region II to region III. Theseoperating points of the transistor are all on the Ids-Vds characteristiccurve where Vgs=Vdd. In other words, the Ids-Vds characteristic curvewhere Vgs=Vdd is only required for determination of the boundarycondition.

The Ids-Vds characteristic curve where Vgs=Vdd is modeled as shown inFIG. 8, using the resistance, Rs, of the resistor 5 and currentparameters I0 and I1.

FIG. 8 shows the result of SPICE simulations when Vgs=Vdd, and a modelrepresenting the result by current parameters.

The source model 6 charges and discharges the output load through theresistor 5 (with the resistance Rs). Thus, the Ids-Vds characteristicsof region III must be expressed by Ids=Vds/Rs. The saturation current atthe boundary point between region II and region III is represented bythe current parameter I1, and the saturation current where Vds=Vdd isrepresented by the current parameter I0.

For this E(t) model, since the saturation current at the boundary pointbetween region II and region III is represented by I1, a region whereVds≦Rs·I1 is referred to as the linear region, and a region whereVds>Rs·I1 as the saturation region. Further, by separately defining thesaturation current where Vds=Vdd as I0, the saturation current in regionII is represented by I0 and I1 as later described, to express thecharacteristics that the drain current Ids decreases gradually withdecreasing drain voltage Vds (or with increasing output terminal voltageV2) in the saturation region.

The values of Δt1, V1, and Δt2 are described using parameters Rs, I0,and I1 which has previously been extracted. Also, as shown in FIG. 9,the relationship between the gate voltage and the current across thetransistor is approximated by a straight line, and a point ofintersection of this straight line and the axis of the gate potential isdefined as Vth0 and registered in the delay-parameter database.

Next, a concrete calculation method for determining the shape of theE(t) waveform is described.

<1. Determination of Δt1>

Firstly, Δt1 is determined. For determination of Δt1, initial values,Δt1=Tslew−t0, V1=E, and Δt2=0, are given. This is equivalent to assumingthat E(t) takes the shape as shown in FIG. 7, and the transistortransitions from the saturation to the linear region whent=Tslew=t0+Δt1.

At this time, when t=Tslew, the PMOS transistor P1 is at the boundarypoint between its saturation and linear regions. As previouslydescribed, the current in this condition is registered as I1 in thedelay-parameter database.

In the description of FIG. 5, the shape of the input waveform is assumedto linearly change from high (potential Vdd) to low (potential of 0 V).Thus, modeling using the current parameters shown in FIG. 8 is also madeassuming that the waveform shape changes linearly.

However, as shown in FIG. 10, actual input waveforms will deviate fromthe linear waveform. FIG. 10 shows one example of actual inputwaveforms. While FIG. 10 shows an example of input waveforms thattransition from low to high, the following description also applies toinput waveforms that transition from high to low.

Now, two points Va and Vb of potential are selected which determine theinclination of the actual input waveform. At a time when a linearwaveform which passes through these two points meets the power supplyvoltage Vdd (or the ground potential GND when the input waveformtransitions from low to high), the rate of the amount of deviation ofthe linear waveform from the actual input waveform to the power supplyvoltage Vdd (this rate is hereinafter referred to as “TinDrop”) ismeasured to form an exponential waveform (with a time constant TinTau)which passes through the point of deviation (the point on the linearwaveform at time Tslew) and a higher one of the potential points used todetermine the inclination of the input waveform (e.g., Va in the exampleof FIG. 10) (or a lower one Vb of the potential points when the inputwaveform transitions from low to high). This waveform can be expressedby only TinDrop and TinTau, which two quantities are thus defined asinput waveform information.

As shown by way of example in FIG. 10, when t=Tslew, the gate voltageVgs takes a value that drops by the amount calculated from Tindrop×E.Thus, actual current when t=Tslew is expressed as d·I1, which is theproduct of the current parameter I1 and a correction factor d expressedas follows:

$\begin{matrix}{d = \frac{{\left( {1 - {TinDrop}} \right)E} - {{Vth}\; 0}}{E - {{Vth}\; 0}}} & (6)\end{matrix}$

Using the equation of V2(t) when Δt2=0, the following equation holdstrue for E(t) where t=Tslew.E(Tslew)=V2(Tslew)+dI1Rs=E  (7)

However, the fact is that the initial values (Δt1=Tslew−t0, V1=E, andΔt2=0) are not always a solution that satisfies equation (7). Thus, theresult can be E(Tslew)>E or E(Tslew)<E.

First, the case where E(Tslew)≧E indicates that t≦Tslew, where t is thetime when the saturation region with varying gate voltage Vgs (region I)transitions to the linear region (region III). More specifically, thecase where E(Tslew)>E indicates that the value of the term,V2(Tslew)+d·I1·Rs, of equation (7) is large. Thus, in order to satisfyequation (7), the current value at the boundary between region I andregion III must be reduced to below I1. Referring to FIG. 5, the stateas shown by the second pattern is required. The second pattern showsthat, when t<Tslew, the transistor transitions from region I to regionIII before the gate voltage Vgs reaches Vdd. In other words, the secondpattern corresponds to the case where the output load capacitance issmall, showing with reference to FIG. 10 that the charging of the outputload is completed before the increase in the gate voltage Vgs reachesthe point of deviation between the linear waveform and the actual inputwaveform. Thus, exponential approximation is not necessary, and theliner waveform determined by the Tslew value can be used as the inputwaveform.

For the E(t) model, E(t0+Δt1)=E when t=t0+Δt1, the time corresponding tothe boundary point between region I and region III. Thus, the followingequation holds true:V2(t0+Δt1)+Rsi(t0+Δt1)=E  (8)

Since we assume that this is the second pattern where Δt2=0, V2(t0+Δt1)can be expressed, using equation (4), as follows:

$\begin{matrix}{{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} = {\left\{ {{\Delta\; t\; 1} - \frac{z}{p_{1}p_{2}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)}} - {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{2}}\Delta\; t\; 1} \right)}}} \right\}\;\frac{E}{\Delta\; t\; 1}}} & (9)\end{matrix}$

Here, i(t0+Δt1) is the current in the saturation region. The currenti(t) is considered as follows: (1) In the simplest transistor model, thedrain current in the saturation region is i∞(Vgs−Vth)², and it is i∞t²when Vgs is approximately expressed as a linear equation; (2) taking thefixed delay t0 into account, t must be t0 and i must be 0; and (3) usinglinear approximation, Vgs=Vdd when t=Tslew, showing the Ids-Vdscharacteristics of FIG. 8. The current value when t=Tslew is the currentvalue I1 at the boundary between the linear and saturation regions whenVgs=Vdd. Thus, i(Tslew)=I1 when t=Tslew. The equation that satisfiesthis condition is as follows:

$\begin{matrix}{{i(t)} = {I\; 1\left( \frac{t - {t\; 0}}{{Tslew} - {t\; 0}} \right)^{2}}} & (10)\end{matrix}$

The current i(t0+Δt1) can be expressed by this equation. Substitutingequations (9) and (10) in equation (8) yields the following relation:

$\begin{matrix}{{E\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} = {{{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} + {\left( \frac{\Delta\; t\; 1}{{Tslew} - {t\; 0}} \right)^{2}{RsI}\; 1}} = E}} & (11)\end{matrix}$

Further, substituting equation (9) for V2(t0+Δt1) and changing theexpression yields the following equation (12):

$\begin{matrix}{{{\left\{ {{- \frac{z}{p_{1}p_{2}}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)}} - {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{2}}\Delta\; t\; 1} \right)}}} \right\} E} + {{RsI}\; 1\;\frac{\Delta\; t\; 1^{3}}{\left( {{Tslew} - {t\; 0}} \right)^{2}}}} = 0} & (12)\end{matrix}$

This equation has one solution when Δt1>0.

Next, when E(Tslew)<E, the saturation current I1 must be increasedconversely, and transistor operation transitions from the saturationregion with varying gate voltage Vgs (region I) to the saturation regionwith fixed gate voltage Vgs (Vgs=E) (region II) and then to the linearregion (region III). That is, the case corresponds to the first patternof the current waveform shown in FIG. 5.

From the fact that Vgs=E at the boundary point between regions I and IIand when considering the initial setting, E(t) reaches this boundarypoint when Tslew=t0+Δt1. When E(t) is expressed as shown in FIG. 6, thevalue of Δt1 is obtained from t0=Tslew−Δt1.

As another exception when solving equation (12), there is the case whereΔt1=Tslew−t0≦0. This case is understood to mean that Vgs=Vdd alreadywhen output current starts to flow, because it takes time to cause theflow of output current. The model in this state is considered toinstantaneously move to the Ids-Vds characteristic curve where Vgs=Vdd.Thus, region I does not exist, and the model starts from region II. Thatis, Δt1=0.

In summary, Δt1 is as follows: (1) In the first case where Tslew−t0≦0,Δt1=0; (2) In the second case where E(Tslew)≧E, since t=t0+Δt1≦Tslew,Δt1≦Tslew−t0 and Δt1 satisfies the following equation:

$\begin{matrix}{{{\left\{ {{- \frac{z}{p_{1}p_{2}}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)}} - {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}{\exp\left( {{- p_{2}}\Delta\; t\; 1} \right)}}} \right\} E} + {{RsI}\; 1\;\frac{\Delta\; t\; 1^{3}}{\left( {{Tslew} - {t\; 0}} \right)^{2}}}} = 0} & (13)\end{matrix}$and (3) In the third case where E(Tslew)<E, Δt1=Tslew−t0.

<2. Determination of V1>

For V1, the following equation also holds true:V2(t0+Δt1)+Rsi(t0+Δt1)=V1  (14)

This equation is used for determining V1.

In the first case where Δt1=0, Vgs=Vdd already when t=t0. Thus, i(t0) ison the Ids-Vds characteristic curve of FIG. 6. Further, since the outputload is not charged by definition of the fixed delay t0 and V2(t0)=0,Vds=Vdd and i(t0) must be I0. Accordingly, V1=Rs·I0. The second casewhere 0<Δt1≦Tslew−t0 corresponds to the second pattern in which region Itransitions to region III, so that V1=E. The third case whereΔt1=Tslew−t0 corresponds to the first pattern, indicating that region Itransitions to region II at time Tslew. Since the output load is beingcharged during time Δt1, V2(t0+Δt1)>0. Further, V2(t0+Δt1)<E−Rs·I1 sincethe current flowing is greater than the drain current Ids at theboundary point between regions II and III.

From the above description, it is found that V2(t0+Δt1) is in thefollowing range: 0<V2(t0+Δt1)<E−Rs·I1. Thus, the operating point of thetransistor is in the position shown in FIG. 11. FIG. 11 shows a portioncorresponding to region II, which is extracted from FIG. 8. Referring toFIG. 11, the current value (saturation current value) in region II,i(t0+Δt1), can be expressed using the current parameters I0 and I1 asfollows:

$\begin{matrix}{{i\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} = {{I\; 0} - {\frac{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)}{E - {{RsI}\; 1}}\left( {{I\; 0} - {I\; 1}} \right)}}} & (15)\end{matrix}$

However, equation (15) is the equation obtained when the duration oftime Δt2 corresponds to the “saturation region with fixed gate voltage”,and the gate voltage Vgs is fixed at Vdd.

The actual input waveform, as shown in FIG. 10, has risen slightlyaccording to the shape of the E(t) waveform. Thus, an error is producedif calculation is made by using the current value i(t0+t1) obtained onthe assumption that Vgs=Vdd.

To reflect this error in the calculations of V1 and Δt2 which is to bedescribed later, the mean gate voltage Vgs during Δt2 is calculated asfollows:

$\begin{matrix}{\overset{\_}{Vgs} = {E\left( {1 - {\frac{{TinDrop} \times {TinTau}}{\Delta\; t\; 2}\left( {1 - {\exp\left( {- \frac{\Delta\; t\; 2}{TinTau}} \right)}} \right)}} \right)}} & (16)\end{matrix}$

Here, when Δt2=0, Vgs=E(1−TinDrop). Thus, when TinDrop=0, Vgs=E.

Using this mean gate voltage Vgs, the current parameters I0 and I1 inthe delay-parameter database are corrected respectively to xI0 and xI1for delay calculation.

Here, the correction factor x can be expressed by the followingequation:

$\begin{matrix}{x = \frac{\overset{\_}{Vgs} - {{Vth}\; 0}}{E - {{Vth}\; 0}}} & (17)\end{matrix}$

Substituting xI0 and xI1 respectively for I0 and I1 and using equation(15) for i(t0+Δt1), we find the value of V1 that satisfies the followingequation:

$\begin{matrix}{{{Et}\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} = {{{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} + {{Rs}\left( {{{xI}\; 0} - {\frac{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)}{E - {{xRsI}\; 1}}\left( {{{xI}\; 0} - {{xI}\; 1}} \right)}} \right)}} = {V\; 1}}} & (18)\end{matrix}$

Further using equation (5), V2(t0+Δt1) can be expressed as:

$\begin{matrix}{{V\; 2\left( {{t\; 0} + {\Delta\; t\; 1}} \right)} = {\left\{ {{\Delta\; t\; 1} - \frac{z}{p_{1}p_{2}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)}} - {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{2}}\;\Delta\; t\; 1} \right)}}} \right\}\frac{V\; 1}{\Delta\; t\; 1}}} & (19)\end{matrix}$

From equations (18) and (19), V1 can be expressed as:

$\begin{matrix}{{V1} = \frac{{xRsI}\; 0}{\begin{matrix}{1 - {\frac{E - {{RsxI}\; 0}}{E - {{RsxI}\; 1}}\left\{ {{\Delta\; t\; 1} - \frac{z}{p_{1}p_{2}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}\;\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)} -} \right.}} \\{\left. {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{2}}\;\Delta\; t\; 1} \right)}} \right\}\frac{1}{\Delta\; t\; 1}}\end{matrix}}} & (20)\end{matrix}$

<3. Determination of Δt2>

Since the second case corresponds to the second pattern, Δt2=0. In thefirst and third cases, even at the boundary between regions II and III,the following equation holds:Et(t0+Δt1+Δt2)=V2(t0+Δt1+Δt2)+xRsI1=E  (21)

Region II is the saturation region where Vgs=Vdd, and at the boundarybetween regions II and III, i(t0+Δt1+Δt2)=x·I1. As the equation fordetermining V2(t0+Δt1+Δt2), equation (3) (in the first case) or equation(5) (in the third case) is used as in the other cases. Then, the valueof Δt2 is determined to satisfy equation (21).

From the above description, we summarize the equations for the values ofΔt1, V1, and Δt2 as follows. In the first case where Tslew−t0≦0, Δt1=0,V1=RsI0, and Δt2 satisfies the equation: V2(t0+Δt1+Δt2)+xRs·I1=E. In thesecond case where 0<Δt1≦Tslew−t0, Δt1 satisfies:

$\begin{matrix}{{{\left\{ {{- \frac{z}{p_{1}p_{2}}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)}} - {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{2}}\Delta\; t\; 1} \right)}}} \right\} E} + {{RsI}\; 1\;\frac{\Delta\; t\; 1^{3}}{\left( {{Tslew} - {t\; 0}} \right)^{2}}}} = 0} & (22)\end{matrix}$and V1=E and Δt2=0.

In the third case where Δt1>Tslew−t0, Δt1=Tslew−t0, V1 can be expressedas:

$\begin{matrix}{{V\; 1} = \frac{{RsxI}\; 0}{\begin{matrix}{1 - {\frac{E - {{RsxI}\; 0}}{E - {{RsxI}\; 1}}\left\{ {{\Delta\; t\; 1} - \frac{z}{p_{1}p_{2}} + {\frac{p_{1} - z}{p_{1}\left( {p_{1} - p_{2}} \right)}\;\exp\left( {{- p_{1}}\Delta\; t\; 1} \right)} -} \right.}} \\{\left. {\frac{p_{2} - z}{p_{2}\left( {p_{1} - p_{2}} \right)}\;{\exp\left( {{- p_{2}}\;\Delta\; t\; 1} \right)}} \right\}\frac{1}{\Delta\; t\; 1}}\end{matrix}}} & (23)\end{matrix}$and Δt2 satisfies the equation: V2(t0+Δt1+Δt2)+Rs×I1=E.

This solution remain unchanged regardless of the rise and fall of thewaveform.

Next, the delay calculation method will be described with reference to aflowchart. FIG. 12 is a flowchart illustrating the flow of the delaycalculation method according to the preferred embodiment of the presentinvention. In step S1, initial conditions are set as follows:Δt1=Tslew−t0, V1=E, and Δt2=0.

In step S2, E(Tslew) is calculated from equation (7). When E(Tslew)≧E,the process goes to step S3 where E(t0+Δt1) is calculated from equation(11). This calculation is repeated with different values of Δt1 untilE(t0+Δt1)=E, thereby to determine the value of Δt1 that satisfiesE(t0+Δt1)=E. Since the case where E(Tslew)≧E corresponds to the secondpattern, the E(t) waveform is as shown in FIG. 7 and can be determinedby determining the value of Δt1.

When E(Tslew)<E, the process goes to step S4, where the mean gatevoltage Vgs is first calculated from equation (16). Then, using the meangate voltage Vgs, the correction factor x is calculated from equation(17). Then, the current parameters I0 and I1 are corrected respectivelyto xI0 and xI1, and V1 is calculated from equation (18).

The process then goes to step S5, where the calculation is repeated withdifferent values of Δt2 until equation (21) holds true. Thereby, thevalue of Δt2 that satisfies equation (21) is determined.

Since the case where E(Tslew)<E corresponds to the first pattern, theE(t) waveform can be determined by determining the values of V1, Δt1,and Δt2.

In step S6, the shape of the E(t) waveform is determined using thevalues of V1, Δt1, and Δt2 obtained in the above steps.

By determining the shape of the E(t) waveform in this way, the waveformof the output terminal voltage V2(t) of each macrocell can be obtained.In step S7, the waveform of the output terminal potential V2(t) iscalculated using E(t). In the case where interconnection is implementedas an RC network, the potential waveform at the end of theinterconnection can also be obtained by analyzing a time response at theend of the interconnection when V2(t) is applied to the output terminalof each cell. In step S8, a delay value and information on the followinginput waveform are calculated from the potential waveform at each node.

FIGS. 13 to 16 show the output waveform (model) calculated using thedelay calculation method according to the preferred embodiment of thepresent invention, and the output waveform obtained through SPICEsimulations. FIGS. 13 and 14 show the cases where the output loadcapacitance is small (C=0.05 pF), and FIGS. 15 and 16 show the caseswhere the output load capacitance is large (C=2.00 pF). For each of thesmall and large output load capacitances, calculations are made for twocases: the case where the waveform shape changes linearly (FIGS. 13 and15); and the case where the waveform shape changes exponentially (FIGS.14 and 16).

As shown in FIGS. 13 to 16, irrespective of any condition, the model andthe SPICE show similar results. As shown in FIGS. 13 and 14, when theoutput load capacitance is small, the output waveform shape does not somuch depend on the input waveform shape. Thus, the output waveforms inFIGS. 13 and 14 are generally similar in shape. This is because, sincethe charging of the output load is completed before the input waveformreaches Vdd, linear approximation of the input waveform leads to a goodresult.

In FIGS. 15 and 16, since the output load capacitance is large, theinput waveform shape for the period before reaching Vdd influences theoutput waveform shape. As a result, the output waveforms in FIGS. 15 and16 give different results.

In FIG. 15, the model output waveform and the SPICE output waveform haveroughly the same shape and overlap each other.

From the above, it is found that the delay calculation method accordingto the preferred embodiment of the present invention allows accuratecalculations of output waveforms by adopting input waveform informationthat gives the shapes of input waveforms. Since output waveforms can becalculated with accuracy, if a time difference between input and outputwaveforms is measured with, for example, 50% of Vdd, delay values can becalculated with a small margin of error.

The present invention appropriately divides, depending on loadconditions, the case where the E(t) waveform can be determined by onlyTslew and the case where the E(t) waveform cannot be determined by onlyTslew. In the latter case, parameters are corrected using a differencein the mean gate voltage Vgs in the saturation region where the gatevoltage is constant. Consequently, even in the case where differentdelay values are calculated from the same input Tslew value, accuratedelay calculations are possible.

Also, using the technique for calculating the correction factor x forparameter correction causes virtually no increase in calculation timewhen compared with that in conventional delay calculation methods.

While the delay calculation method according to the preferred embodimentof the present invention employs a voltage source model that representstransistor operation by determining the E(t) waveform of the powersupply, this method is also applicable to other delay calculationmethods such as that employing a current source model. In the case ofthe current source model, instead of determining the E(t) waveform, acurrent source is set so as to directly output current waveforms asshown in FIG. 5. Even in this model's case, according to the same idea(that the mean gate voltage Vgs is calculated for parameter correction),it is possible to calculate saturation current (current in region II)with accuracy, and the use of such saturation current for calculationincreases the accuracy of calculation.

While the invention has been shown and described in detail, theforegoing description is in all aspects illustrative and notrestrictive. It is therefore understood that numerous modifications andvariations can be devised without departing from the scope of theinvention.

1. A delay calculation method for calculating a delay in a logic circuitincluding at least a MOS transistor, the delay calculation methodcomprising the steps of: modeling the transistor using: a resistanceelement whose resistance value is fixed, and a power supply whosevoltage varies with time; determining a shape of a voltage waveform ofsaid power supply based on input waveform information associated with ashape of an input waveform of said logic circuit; calculating, by usinga machine, a delay in the logic circuit; and displaying or storing oroutputting the calculated delay, wherein E(t) represents voltagewaveform of said power supply, and wherein determining E(t) includes thesteps of: (a) determining whether E(Tslew) is greater than or equal toE, wherein Tslew represents the time when the input waveform hasfinished its transition to 0 or Vdd and E represents a first constantvoltage, (b) upon a determination that E(Tslew) is not greater than orequal to E, calculating a mean gate voltage, (c) using the mean gatevoltage to calculate V1, wherein V1 represents a second constantvoltage, (d) determining whether E(t0+Δt1+Δt2) is equal to E, wherein t0defines a fixed delay time required for the input waveform to exceed athreshold voltage of the transistor, Δt1 defines the time required forE(t) to increase from 0 to V1, and Δt2 defines the time required forE(t) to increase from V1 to E, and (e) upon a determination thatE(t0+Δt1+Δt2) is equal to E, determining the waveform shape of E(t). 2.The delay calculation method according to claim 1, wherein said inputwaveform information includes: information on potential of said inputwaveform at a time when a linear waveform that requires the same amountof time as said input waveform to transition between predetermined twopotentials meets a power supply voltage or ground voltage which dependson whether said input waveform transitions from low to high or from highto low, respectively; and information on time constant of an exponentialwaveform which passes through said potential of said input waveform atsaid time and a higher or lower one of said predetermined two potentialswhich depends on whether said input waveform transitions from low tohigh or from high to low, respectively.
 3. The delay calculation methodaccording to claim 1, wherein operating characteristics of saidtransistor being modeled are represented as divided into a first regionwhere current changes with changing gate potential, a second regionwhere current decreases gradually while gate potential is constant, anda third region where current decreases while gate potential is constant,said second region corresponding to a saturation region of saidtransistor, said third region corresponding to a linear region of saidtransistor; and when said transistor transitions from said first regionto said second region and then to said third region, the shape of saidvoltage waveform of said power supply is determined according to saidinput waveform information.
 4. The delay calculation method according toclaim 3, wherein a saturation current value of said transistor in saidsecond region is calculated based on said input waveform information andis used to determine the shape of said voltage waveform of said powersupply.
 5. The delay calculation method according to claim 4, whereinsaid saturation current value is calculated according to a mean gatepotential of said transistor in said second region.